Dualities and models in various dimensions

Working within the path-integral framework we first establish a duality between the partition functions of two $U\left(1\right)$ gauge theories with a theta term in $d=4$ space-time dimensions. Then, after a dimensional reduction to $d=3$ dimensions we arrive to the partition function of a $U\left(1\right)$ gauge theory coupled to a scalar field with an action that exhibits a Dirac monopole solution. A subsequent reduction to $d=2$ dimensions leads to the partition function of a theory in which the gauge field decouples from two scalars which have non-trivial vortex-like solutions. Finally this $d=2$ partition function can be related to the bosonized version of the two-dimensional QED₂ (Schwinger) model.